Question

A Uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends, If the tension generated in the rod due to rotation is $T\left(x\right)$ at a distance $x$ from the axis, then which of the following graphs depict it closely ?

• A.
• B.
• C.
• D.

Answer 1

The correct option is D



Consider a small element of length $dx$ of mass $dm$ at a distance of $x$ from the axis of the rotation.

Here, Tension acts as the centripetal force,

$\Rightarrow dT=\left(dm\right)x{\omega }^2$

Then, the tension along the rod is given by,

$T={\int }_0^{T}dT={\int }_0^x\left(dm\right){\omega }^{2}x$

$\Rightarrow T={\int }_0^x\left(\frac{m}{l}dx\right){\omega }^{2}x$

$\text{or}T=\frac{m{\omega }^2}{l}(l^2-x^2)$

It is a parabola between $T$ and $x$

Hence, option (B) is the correct answer.